well most likely there would be the magnetic field and the electric field from the potential added.
Normally the electric field lines are perpendicular to the magnetic ones.
The way the field lines would go out also depends on the geometry of the magnet.
although I'm not sure what happens wen you would switch that magnet into a circuit aka make the magnet as part of a conductor running a current through it , because magnetism is due to the electrons ligned up but when current runs might line up differently but im not sure someone else has to comment on this one.
I think it is not so simple to say what happens, but here is my guess. For ferromagnets the net field is due to a large enough number of the magnetic domains in the material becoming aligned, but below the Curie temperature all the domains will be spontaneously magnetised, whether the net field is strong or not. The interactions which cause this spontaneous ordering are really quite strong, so I don't think a current flowing through the material would be enough to mess it up unless perhaps it was a super-strong current (though I don't know).
To clarify further; in many ferromagnetic materials I believe the conduction electrons are indeed the ones responsible for the magnetisation, since the outer shell electrons tend to be the ones that end up in the conduction band, and it is these electrons that cause the magnetisation (the inner shell ones are all paired up and cancel out each others magnetic dipoles). These electrons migrate through the material when a voltage is applied, but I expect that within a domain they interact to (on average) line up all their dipoles regardless of their net movement. So I predict that there will be no measurable change in the net magnetic field when such a current is applied :p. Maybe you will notice something if you use a very sensitive magnetometer though.